1.8 Solving systems of linear equations and polynomial equations with technologyIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   At a concert, adult tickets cost $a$ USD each and child tickets cost $c$ USD each. Three adults and two children pay 98 USD in total. Two adults and five children pay 124 USD in total.

   (a)

   Which pair of equations models the situation?

   [1 mark]

   • A$3a+2c=124$ and $2a+5c=98$
   • B$3a+2c=98$ and $2a+5c=124$
   • C$3a+2c=98$ and $5a+2c=124$
   • D$a+c=98$ and $a+c=124$

   (b)

   Use your GDC to solve the equations. What is the price of one child ticket?

   [1 mark]

   • A22 USD
   • B14 USD
   • C18 USD
   • D16 USD

   (c)

   Find the total cost of four adult tickets and three child tickets.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A rectangular photo frame has a length that is 5 cm greater than its width, $x$ cm. The area of the frame is $130\ \mathrm{cm^{2}}$.

   (a)

   Which equation is satisfied by $x$?

   [1 mark]

   • A$x^{2}+5x-130=0$
   • B$x^{2}+5x+130=0$
   • C$2x+5=130$
   • D$x^{2}-5x-130=0$

   (b)

   A student solves the equation using a GDC and finds the roots $x\approx9.17$ and $x\approx-14.2$. Which statement is correct?

   [1 mark]

   • ABoth roots give valid widths.
   • BOnly $x=-14.2$ gives a valid width.
   • COnly $x=9.17$ gives a valid width, because a width cannot be negative.
   • DNeither root gives a valid width.

   (c)

   Find the perimeter of the frame, correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A school shop sells notebooks, pens and markers. Let their prices be $x$, $y$ and $z$ AED respectively. Order 1: one notebook, one pen and one marker cost 10 AED. Order 2: two notebooks, three pens and one marker cost 17 AED. Order 3: three notebooks, one pen and four markers cost 31 AED.

   (a)

   Write down a system of three linear equations in $x$, $y$ and $z$ that models the three orders.

   [3 marks]

   (b)

   Use your GDC to solve the system, and hence find the cost of two notebooks, four pens and one marker.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The height $h$ metres, above the ground at the foot of a cliff, of a stone thrown from the top of the cliff is modelled by $h=at^{2}+bt+c$, where $t$ is the time in seconds after it is thrown. The stone's height is $22.1$ m when $t=1$, $22.4$ m when $t=2$ and $12.9$ m when $t=3$.

   (a)

   (i) Write down three equations in $a$, $b$ and $c$.
   (ii) Use your GDC to find the values of $a$, $b$ and $c$, and state what $c$ represents.

   [6 marks]

   (b)

   Use the model $h=-4.9t^{2}+15t+12$.
   (i) Find the time at which the stone hits the ground.
   (ii) Find the length of time for which the stone is more than $20$ m above the ground.

   [6 marks]

   Total for question 4: 12 marks

End of questions